Noname manuscript No. (will be inserted by the editor) Gravity, antimatter and the Dirac-Milne universe Gabriel Chardin · Giovanni Manfredi Received: date / Accepted: date Abstract We review the main arguments against antigravity, a different ac- celeration of antimatter relative to matter in a gravitational field, discussing and challenging Morrison's, Good's and Schiff's arguments. Following Price, we show that, very surprisingly, the usual expression of the Equivalence Prin- ciple is violated by General Relativity when particles of negative mass are supposed to exist, which may provide a fundamental explanation of MOND phenomenology, obviating the need for Dark Matter. Motivated by the observation of repulsive gravity under the form of Dark Energy, and by the fact that our universe looks very similar to a coasting (nei- ther decelerating nor accelerating) universe, we study the Dirac-Milne cosmol- ogy, a symmetric matter-antimatter cosmology where antiparticles have the same gravitational properties as holes in a semiconductor. Noting the similar- ities with our universe (age, SN1a luminosity distance, nucleosynthesis, CMB angular scale), we focus our attention on structure formation mechanisms, finding strong similarities with our universe. Additional tests of the Dirac-Milne cosmology are briefly reviewed, and we finally note that a crucial test of the Dirac-Milne cosmology will be soon realized at CERN next to the ELENA antiproton decelerator, possibly as early as fall 2018, with the AEgIS, ALPHA-g and Gbar antihydrogen gravity experiments. Keywords Antimatter · Gravity · Cosmology · Dark Energy · Equivalence principle G. Chardin CNRS, 3 rue Michel-Ange, F-75016 Paris, France E-mail: [email protected] arXiv:1807.11198v1 [gr-qc] 30 Jul 2018 G. Manfredi Universit´ede Strasbourg, CNRS, IPCMS UMR 7504, F-67000 Strasbourg, France. E-mail: [email protected] 2 Gabriel Chardin, Giovanni Manfredi 1 Introduction The vast majority of theoretical physicists believe that, if a difference in accel- eration between matter and antimatter exists, it can only be extremely small. Few consider possible an antigravity where antihydrogen would \fall up", as the CERN presents the three current experiments testing the equivalence prin- ciple by the ELENA antiproton decelerator [1]. Why then these three experiments, AEgIS [2], ALPHA-g [3] and Gbar [4], aim only at a precision of the order of one percent, at least in a first stage, while at the same time, the BASE experiment [5] claims to impose constraints on any anomalous gravity for antimatter at the sub-ppm level? Could it be that antigravity, in the sense of antimatter \falling up", is actually a prediction, which seems at first antinomic, of general relativity? In a first part of this work, we briefly review the impossibility arguments against antigravity, focusing on Schiff's [6], Morrison's [7] and Good's [8] argu- ments, showing why they are probably ineffective. We then discuss, also rather briefly, the so-called Klein paradox, or vacuum polarization, which provides some elements of answer concerning the impossibility of negative energy states and negative mass. More fundamentally, we describe the argument by Price [9] showing that general relativity violates maximally the usual expression of the Equivalence Principle as soon as the existence of negative mass, possibly as virtual constituents of the quantum vacuum, is allowed. This will lead us to the Dirac-Milne universe [10], and a possible explana- tion of the repulsive gravity that we observe in cosmology, called Dark Energy for lack of better comprehension. This matter-antimatter universe is impres- sively concordant, and has also, a fact that is often not realized, a simple physical analog with the electron-hole system in a semiconductor. Next, we discuss the mechanism of structure formation in the Dirac-Milne universe, radically different from that of the Lambda-CDM universe, and show that, without any free parameter, it reproduces several of the features observed in large surveys such as SDSS [11]. In a final part, we discuss the additional experiments and studies that can be realized in the near future to test the Dirac-Milne cosmology. 2 Impossibility arguments Over the years, several impossibility arguments have been raised against anti- gravity. A rather thorough discussion of the main impossibility arguments can be found in the review by Nieto and Goldman [12], dating back to 1991 but still mostly valid today. Truly enough, as soon as we express general relativity as a metric theory, with a single metric, it is difficult to see how gravity could distinguish matter from antimatter since according to the very formalisme of a single metric, all particles must follow the same trajectory. Still we will see that General Relativity does predict gross violations of the Equivalence Prin- ciple as soon as negative mass components are allowed. Also, as was noted by Gravity, antimatter and the Dirac-Milne universe 3 Fig. 1 The particle-antiparticle loop of the Morrison argument. The pair starts at point A and then is risen vertically through the gravitational field to point B, where the pair annihilates into two photons. These two pho- tons are propagated down (dur- ing which they gain energy) to point A. There they reconvert into a pair, but with an extra photon of energy 2meg∆z, where ∆z is the vertical distance be- tween A and B. physicists in solid-state physics [13] and in structure formation [14,15], there could be other expressions of the Equivalence Principle respecting the spirit of General Relativity but violating maximally its usual expression. Coming back to the impossibility arguments, we can summarize them in three classes, that we might call the Morrison argument [7], the Schiff argument [6], and the Good argument [8]. 2.1 Morrison's argument As early as 1958, Morrison, in a celebrated paper associated with his Richtmyer memorial lecture [7], studied the consequences of antigravity in a gedanken experiment that can be summarized in Fig. 1. Basically, the argument states that, if we accept antigravity, energy is not conserved and/or the vacuum becomes unstable. The question(s) that Morrison did not ask was : \Unstable, by how much, and what is the characteristic timescale?" Noting that in some other situations, the vacuum of gravitational structures such as black holes is unstable, since black holes evaporate, one of us tried to estimate this instability at the beginning of the 1990s [16]. Remarkably, the instability that we can expect from antigravity is the same as the Hawking evaporation of black holes, and leads to a temperature of radiation of: kBT ≈ ¯hg=2πc where g is the usual surface gravity. So the answer to Morrison's argument might well be that the instability associated to antigravity is acceptable since it occurs in most situations at an unnoticeably feeble rate, and is observed in other similar situations such as black hole evaporation. Note, for example, that for a black hole of the mass of our Sun, the evaporation timescale is of the order of 2 × 1066 years. As we will see, it is mostly in strong fields, such as those occurring near the horizon of black holes, that this antigravity will lead to significant effects. We will come 4 Gabriel Chardin, Giovanni Manfredi back to this point as such a vacuum polarization is a prediction of general relativity as soon as we allow the existence of negative mass objects in the vacuum. Note that some authors have argued, notably in supersymmetric theories, that antigravity, meaning here a slightly different acceleration of antimatter with respect to matter in a gravitational field, is possible without implying any dissipation or instability. This is, in particular, what Jo¨elScherk proposed in 1979 in another celebrated paper [17] adequately titled \Antigravity, a crazy idea?" Scherk argued that in this case, if we assume the existence of gravivector components in N = 2; 8 supersymmetry, we might expect antimatter to fall slightly more rapidly than matter. Still, in the following, we will assume that Morrison's argument was correct, and that the instability is characterized by Hawking's evaporation temperature and therefore acceptable. 2.2 Schiff's argument In his lectures on gravitation [18], with his second lecture dealing with anti- gravity, John Bell estimated that probably the most stringent impossibility argument against antigravity was the argument brought by Leonard Schiff at the beginning of the 1960s. The Schiff argument [6] can be summarized in the following way: if antimatter antigravitates, then, depending on the composi- tion of the body used to test the equivalence principle, for example beryllium and uranium, two elements with rather different binding energies, these bod- ies should follow different trajectories in gravitational fields. To quantify his statement, Schiff tried to estimate the contribution of the virtual electron- positron pairs, and even more importantly the contribution of the virtual quark-antiquark pairs, arguing that pure antigravity would induce variations of the order of these contributions in the acceleration of bodies with different binding energies. Nieto and Goldman had already noted [12] that Schiff's calculation was incorrect as his calculation did not take correctly into account the infinities arising in the renormalisation procedure. Still, we know that, to an outstand- ing precision, all material bodies, independently of their composition, follow the same trajectories {at least for matter{ for given initial conditions in a grav- itational field. The most precise tests have been provided by the experiments of the E¨otwash group [19] and more recently by the Microscope satellite, the latter providing the most stringent constraint at the level of 2 × 10−14 in its first analysis [20]. Even before the Microscope result, with the Schiff argument in mind, it has been estimated (see e.g. [21]) that antigravity is constrained at the level of one part per billion.